Artian Completed-Event Hamiltonian Theorem: Primitive Direction, Magnitude No-Go, and the Record-Spend Closure Gate

When one completed quantum event fixes a Hamiltonian direction but not its strength Version: 1.0 Concept DOI: 10.5281/zenodo.21428033 Author: Ali Attar Website: quantumtraction.org Main Book anchor: Quantum Traction Theory: Main Book v10.01 Take one homogeneous edge, one quantum of support, equal endpoint labels, and a self-adjoint real-\(J\) generator. Endpoint symmetry leaves exactly one physical reversible direction after the common identity phase is removed: \[ { Comm(S_e)ₛₐ/RI RS_e } \] The direction is unique. Its magnitude is not. The complete local premise set admits the continuum \[ { HC_Hᵗᵒᵗ = E_* [ (1-C_H/2)I + C_H/2S_e ], spec\,HC_Hᵗᵒᵗ = \{(1-C_H)E_*,E_*\}, 0< C_H≤1 } \] Every member carries the same completed upper state and obeys the same finite spectral ceiling. The printed local premises therefore do not select \(C_H=1\). The exact coherent readout makes the event meaning testable: \[ { H_e=Ω_e/2S_e, pu→ v(T) = sin^2\!(Ω_eT/2) } \] At \(C_H=Ω_e t_A=1\), one tick gives \(pu→ v=0.229848847065930…\), not an exact swap. An orthogonal unitary transfer requires at least \[ { ∫ osc(H(T))\, dT≥π } \] The missing magnitude theorem is therefore an allocation theorem. Event records must first be grouped by which funded transaction they represent: \[ { PE = RE/\!~fund } \] If that quotient has one non-null funded class, if the class is primitive reversible transport, and if its action is faithfully represented by shift-invariant spectral contrast, then the closure gate is immediate: \[ { t_A\, osc \!( PₚᵣᵢₘHEPₚᵣᵢₘ ) = C_H=1 } \] The paper also proves coherent/dissipative type separation, distinguishes conductance from Hamiltonian frequency, gives the exact three-address counterexample, defines the finite no-double-spend network polytope, and proves that continuous-time nearest-neighbor hopping does not have exact compact support at positive time. Machine audit: 500 endpoint-commutant trials, 500 phase-gauge trials, 1,000 legal coefficient countermodels, 300 three-site coherent trials, 500 spectral non-additivity trials, 400 network packets, graph distances 1 through 7, and 14/14 detected mutations. Headline status: SIGMA-CEH-PRIMITIVE-HAMILTONIAN-DIRECTION-UNIQUE-ON-DECLARED-CLASS - GREEN SIGMA-CEH-LOCAL-COEFFICIENT-NONIDENTIFIABILITY-NOGO-CLOSED - GREEN SIGMA-CEH-ONE-TICK-EXACT-UNITARY-RELOCATION-NOGO-CLOSED - GREEN SIGMA-CEH-RECORD-SPEND-QUOTIENT-PENDING - AMBER SIGMA-CEH-C-H-EQUALS-ONE-CONDITIONAL-ON-CEAC-SISAE - AMBER Related QTT anchors: Artian Gravity Reference Framework Artian Hamiltonian Framework Artian Lagrangian Framework QTT Derivation of the Principle of Least Action Gravity Authority Derivation Atlas Included public files: the PDF paper and one full reconstruction package containing LaTeX sources, verifier, machine receipts, scientific audit, render audit, metadata, and SHA-256 manifest.